# Waldspurger formula

In representation theory of mathematics, the Waldspurger formula relates the special values of two L-functions of two related admissible irreducible representations. Let k be the base field, f be an automorphic form over k, π be the representation associated via the Jacquet–Langlands correspondence with f. Goro Shimura (1976) proved this formula, when $k=\mathbb {Q}$ and f is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper. Marie-France Vignéras (1980) proved this formula, when $k=\mathbb {Q}$ and f is a newform. Jean-Loup Waldspurger, for whom the formula is named, reproved and generalized the result of Vignéras in 1985 via a totally different method which was widely used thereafter by mathematicians to prove similar formulas.

## Statement

Let $k$ be a number field, $\mathbb {A}$ be its adele ring, $k^{\times }$ be the subgroup of invertible elements of $k$, $\mathbb {A} ^{\times }$ be the subgroup of the invertible elements of $\mathbb {A}$, $\chi ,\chi _{1},\chi _{2}$ be three quadratic characters over $\mathbb {A} ^{\times }/k^{\times }$, $G=SL_{2}(k)$, ${\mathcal {A}}(G)$ be the space of all cusp forms over $G(k)\backslash G(\mathbb {A} )$, ${\mathcal {H}}$ be the Hecke algebra of $G(\mathbb {A} )$. Assume that, $\pi$ is an admissible irreducible representation from $G(\mathbb {A} )$ to ${\mathcal {A}}(G)$, the central character of π is trivial, $\pi _{\nu }\sim \pi [h_{\nu }]$ when $\nu$ is an archimedean place, ${A}$ is a subspace of ${{\mathcal {A}}(G)}$ such that $\pi |_{\mathcal {H}}:{\mathcal {H}}\to A$. We suppose further that, $\varepsilon (\pi \otimes \chi ,1/2)$ is the Langlands $\varepsilon$-constant [ (Langlands 1970); (Deligne 1972) ] associated to $\pi$ and $\chi$ at $s=1/2$. There is a ${\gamma \in k^{\times }}$ such that $k(\chi )=k({\sqrt {\gamma }})$.
Definition 1. The Legendre symbol $\left({\frac {\chi }{\pi }}\right)=\varepsilon (\pi \otimes \chi ,1/2)\cdot \varepsilon (\pi ,1/2)\cdot \chi (-1).$
* Comment. Because all the terms in the right either have value +1, or have value −1, the term in the left can only take value in the set {+1, −1}.
Definition 2. Let ${D_{\chi }}$ be the discriminant of $\chi$. $p(\chi )=D_{\chi }^{1/2}\sum _{\nu {\text{ archimedean}}}\left\vert \gamma _{\nu }\right\vert _{\nu }^{h_{\nu }/2}.$
Definition 3. Let $f_{0},f_{1}\in A$. $b(f_{0},f_{1})=\int _{x\in k^{\times }}f_{0}(x)\cdot {\overline {f_{1}(x)}}\,dx.$
Definition 4. Let ${T}$ be a maximal torus of ${G}$, ${Z}$ be the center of ${G}$, $\varphi \in A$. $\beta (\varphi ,T)=\int _{t\in Z\backslash T}b(\pi (t)\varphi ,\varphi )\,dt.$
* Comment. It is not obvious though, that the function $\beta$ is a generalization of the Gauss sum.
Let $K$ be a field such that $k(\pi )\subset K\subset \mathbb {C}$. One can choose a K-subspace${A^{0}}$ of $A$ such that (i) $A=A^{0}\otimes _{K}\mathbb {C}$; (ii) $(A^{0})^{\pi (G)}=A^{0}$. De facto, there is only one such $A^{0}$ modulo homothety. Let $T_{1},T_{2}$ be two maximal tori of $G$ such that $\chi _{T_{1}}=\chi _{1}$ and $\chi _{T_{2}}=\chi _{2}$. We can choose two elements $\varphi _{1},\varphi _{2}$ of $A^{0}$ such that $\beta (\varphi _{1},T_{1})\neq 0$ and $\beta (\varphi _{2},T_{2})\neq 0$.
Definition 5. Let $D_{1},D_{2}$ be the discriminants of $\chi _{1},\chi _{2}$.
    $p(\pi ,\chi _{1},\chi _{2})=D_{1}^{-1/2}D_{2}^{1/2}L(\chi _{1},1)^{-1}L(\chi _{2},1)L(\pi \otimes \chi _{1},1/2)L(\pi \otimes \chi _{2},1/2)^{-1}\beta (\varphi _{1},T_{1})^{-1}\beta (\varphi _{2},T_{2}).$
* Comment. When the $\chi _{1}=\chi _{2}$, the right hand side of Definition 5 becomes trivial.
We take $\Sigma _{f}$ to be the set {all the finite $k$-places $\nu \mid \ \pi _{\nu }$ doesn't map non-zero vectors invariant under the action of ${GL_{2}(k_{\nu })}$ to zero}, ${\Sigma _{s}}$ to be the set of (all $k$-places $\nu \mid \nu$ is real, or finite and special).
Theorem — Let $k=\mathbb {Q}$. We assume that, (i) $L(\pi \otimes \chi _{2},1/2)\neq 0$; (ii) for $\nu \in \Sigma _{s}$, $\left({\frac {\chi _{1,\nu }}{\pi _{\nu }}}\right)=\left({\frac {\chi _{2,\nu }}{\pi _{\nu }}}\right)$ . Then, there is a constant ${q\in \mathbb {Q} (\pi )}$ such that $L(\pi \otimes \chi _{1},1/2)L(\pi \otimes \chi _{2},1/2)^{-1}=qp(\chi _{1})p(\chi _{2})^{-1}\prod _{\nu \in \Sigma _{f}}p(\pi _{\nu },\chi _{1,\nu },\chi _{2,\nu })$
Comments:
* The formula in the theorem is the well-known Waldspurger formula. It is of global-local nature, in the left with a global part, in the right with a local part. By 2017, mathematicians often call it the classic Waldspurger's formula.
* It is worthwhile to notice that, when the two characters are equal, the formula can be greatly simplified.
* [ (Waldspurger 1985), Thm 6, p. 241 ] When one of the two characters is ${1}$, Waldspurger's formula becomes much more simple. Without loss of generality, we can assume that, $\chi _{1}=\chi$ and $\chi _{2}=1$. Then, there is an element ${q\in \mathbb {Q} (\pi )}$ such that $L(\pi \otimes \chi ,1/2)L(\pi ,1/2)^{-1}=qD_{\chi }^{1/2}.$

## The case when F$_p$(T) and φ is a metaplectic cusp form

Let p be prime number, $\mathbb {F} _{p}$ be the field with p elements, $R=\mathbb {F} _{p}[T],k=\mathbb {F} _{p}(T),k_{\infty }=\mathbb {F} _{p}((T^{-1})),o_{\infty }$ be the integer ring of $k_{\infty },{\mathcal {H}}=PGL_{2}(k_{\infty })/PGL_{2}(o_{\infty }),\Gamma =PGL_{2}(R)$. Assume that, $N,D\in R$, D is squarefree of even degree and coprime to N, the prime factorization of $N$ is ∏$_ℓ$ℓ$^{αℓ}${\textstyle \prod _{\ell }\ell ^{\alpha _{\ell }}}. We take $\Gamma _{0}(N)$ to the set {(abcd)∈Γ∣c≡0modN}, {\textstyle \left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \Gamma \mid c\equiv 0{\bmod {N}}\right\}, } $S_{0}(\Gamma _{0}(N))$ to be the set of all cusp forms of level N and depth 0. Suppose that, $\varphi ,\varphi _{1},\varphi _{2}\in S_{0}(\Gamma _{0}(N))$.
Definition 1. Let $\left({\frac {c}{d}}\right)$ be the Legendre symbol of c modulo d, ${\widetilde {SL}}_{2}(k_{\infty })=Mp_{2}(k_{\infty })$. Metaplectic morphism $\eta :SL_{2}(R)\to {\widetilde {SL}}_{2}(k_{\infty }),{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\mapsto \left({\begin{pmatrix}a&b\\c&d\end{pmatrix}},\left({\frac {c}{d}}\right)\right).$
Definition 2. Let $z=x+iy\in {\mathcal {H}},d\mu ={\frac {dx\,dy}{\left\vert y\right\vert ^{2}}}$. Petersson inner product $\langle \varphi _{1},\varphi _{2}\rangle =[\Gamma :\Gamma _{0}(N)]^{-1}\int _{\Gamma _{0}(N)\backslash {\mathcal {H}}}\varphi _{1}(z){\overline {\varphi _{2}(z)}}\,d\mu .$
Definition 3. Let $n,P\in R$. Gauss sum $G_{n}(P)=\sum _{r\in R/PR}\left({\frac {r}{P}}\right)e(rnT^{2}).$
Let $\lambda _{\infty ,\varphi }$ be the Laplace eigenvalue of $\varphi$. There is a constant $\theta \in \mathbb {R}$ such that $\lambda _{\infty ,\varphi }={\frac {e^{-i\theta }+e^{i\theta }}{\sqrt {p}}}.$
Definition 4. Assume that $v_{\infty }(a/b)=\deg(a)-\deg(b),\nu =v_{\infty }(y)$. Whittaker function $W_{0,i\theta }(y)={\begin{cases}{\frac {\sqrt {p}}{e^{i\theta }-e^{-i\theta }}}\left[\left({\frac {e^{i\theta }}{\sqrt {p}}}\right)^{\nu -1}-\left({\frac {e^{-i\theta }}{\sqrt {p}}}\right)^{\nu -1}\right],&{\text{when }}\nu \geq 2;\\0,&{\text{otherwise}}.\end{cases}}$
Definition 5. Fourier–Whittaker expansion $\varphi (z)=\sum _{r\in R}\omega _{\varphi }(r)e(rxT^{2})W_{0,i\theta }(y).$ One calls $\omega _{\varphi }(r)$ the Fourier–Whittaker coefficients of $\varphi$.
Definition 6. Atkin–Lehner operator $W_{\alpha _{\ell }}={\begin{pmatrix}\ell ^{\alpha _{\ell }}&b\\N&\ell ^{\alpha _{\ell }}d\end{pmatrix}}$ with $\ell ^{2\alpha _{\ell }}d-bN=\ell ^{\alpha _{\ell }}.$
Definition 7. Assume that, $\varphi$ is a Hecke eigenform. Atkin–Lehner eigenvalue $w_{\alpha _{\ell },\varphi }={\frac {\varphi (W_{\alpha _{\ell }}z)}{\varphi (z)}}$ with $w_{\alpha _{\ell },\varphi }=\pm 1.$
Definition 8. $L(\varphi ,s)=\sum _{r\in R\backslash \{0\}}{\frac {\omega _{\varphi }(r)}{\left\vert r\right\vert _{p}^{s}}}.$
Let ${\widetilde {S}}_{0}({\widetilde {\Gamma }}_{0}(N))$ be the metaplectic version of $S_{0}(\Gamma _{0}(N))$, $\{E_{1},\ldots ,E_{d}\}$ be a nice Hecke eigenbasis for ${\widetilde {S}}_{0}({\widetilde {\Gamma }}_{0}(N))$ with respect to the Petersson inner product. We note the Shimura correspondence by $\operatorname {Sh} .$
Theorem [ (Altug & Tsimerman 2010), Thm 5.1, p. 60 ]. Suppose that K$_φ$=1/√p(√p−e$^{−iθ}$)(√p−e$^{iθ}$) {\textstyle K_{\varphi }={\frac {1}{{\sqrt {p}}\left({\sqrt {p}}-e^{-i\theta }\right)\left({\sqrt {p}}-e^{i\theta }\right)}}}, $\chi _{D}$ is a quadratic character with $\Delta (\chi _{D})=D$. Then $\sum _{\operatorname {Sh} (E_{i})=\varphi }\left\vert \omega _{E_{i}}(D)\right\vert _{p}^{2}={\frac {K_{\varphi }G_{1}(D)\left\vert D\right\vert _{p}^{-3/2}}{\langle \varphi ,\varphi \rangle }}L(\varphi \otimes \chi _{D},1/2)\prod _{\ell }\left(1+\left({\frac {\ell ^{\alpha _{\ell }}}{D}}\right)w_{\alpha _{\ell },\varphi }\right).$